THE 3-MODULAR DECOMPOSITION NUMBERS OF THE
SPORADIC SIMPLE SUZUKI GROUP
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Abstract. The purpose of this paper is to determine the 3-modular decomposition numbers for the sporadic simple Suzuki group Suz. The results are
obtained by a combined use of character theory, explicit construction of modules, and condensation techniques with help of the computer algebra systems
MOC, MeatAxe, TensorCondense, and GAP.
1. Introduction
In the last years many of the decomposition matrices for the sporadic simple
Suzuki group Suz have been calculated. The Brauer trees for all primes and all
blocks of cyclic defect have been determined by G. Hiss and K. Lux [6]. The 2modular and 5-modular decomposition numbers are also known, although proofs
have not yet been published. In this paper we determine the 3-modular decomposition numbers for Suz. Our results are collected in Section 2. Especially, it turns
out that the decomposition matrix for the principal block of Suz is not of wedge
shape, i. e., there are no permutations of columns and rows such that the resulting
matrix is lower triangular. To obtain our results we make substantial use of the
computer algebra systems MOC [5], MeatAxe [13], [14], and GAP [16].
The MOC system has been developed for calculations with ordinary and Brauer
characters, details can be found in [5]. Although a huge amount of data is produced
in the course of our computations with MOC, using the protocol MOC provides
about its actions we are able to give explicit proofs of our results, which can in
principle be checked by hand. Most of the details presented in Sections 4 and 5 are
obtained this way. We assume the reader to be acquainted with the notion of basic
sets of projective characters and of Brauer characters, as well as with the notion
of projective atoms and Brauer atoms. The strategy of the proof is as follows.
We start with a basic set of Brauer characters consisting of ordinary characters restricted to 3-regular classes and a basic set of projective characters. These are then
modified in a series of steps, using linear dependencies amongst Brauer characters
and projective characters, respectively, finally giving a best possible approximation
to the irreducible Brauer characters and the projective indecomposable characters.
We use the matrix of scalar products between Brauer characters and projective
characters as the leading theme in the description of the single steps, as it contains
the necessary information in a most comprised form and is of wedge shape very
soon.
Projective characters are obtained by induction of projective indecomposable
characters of a few maximal subgroups. The necessary results for these subgroups
Date: November 12, 2002.
1991 Mathematics Subject Classification. 20C20, 20C34, 20C40.
1
2
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
are restated in Section 3. As the irreducible Brauer characters of Suz exhaust
all of the irreducible Brauer characters of the triple cover 3. Suz and there is a
bijection between the 3-blocks of Suz and those of 3. Suz, see [4], Lemma V.4.5., it
is customary to use the faithful ordinary characters of 3. Suz to find a suitable basic
set of Brauer characters to begin with and to find relations to refine the current basic
sets. In the sequel, we use faithful and non-faithful ordinary characters without
further notice.
In addition to these purely character theoretic methods, we also make use of the
Benson-Carlson Theorem, see [2]. These methods are sufficient then to complete
the proof for the second block, whereas for the principal block a few questions
remain open. This is where explicit construction and analysis of modules comes
into play. The proof for the principal block is completed in Sections 7 and 8 by an
analysis of several matrix and permutation representations. We make use of the
Aachen implementation of the MeatAxe [14] which also includes the Lattice package
[10] for the computation of submodule lattices. We assume the reader to be familiar
with the standard MeatAxe techniques and the ideas introduced in [10]. We also
apply fixed point condensation, which is described briefly in Section 6. Especially,
we make use of the TensorCondense package [17], [11], the idea of which has already
been described in [12], and which in the meantime has been incorporated into the
MeatAxe [14].
All the ordinary and Brauer character tables we use can be found in [3] and
[8] as far as ordinary tables and Brauer tables are concerned, respectively. These
tables can also be accessed in the computer algebra system GAP. We follow the
enumeration of characters, blocks, and conjugacy classes given in [3] and [8], which
is the same as is provided by GAP. We also use the algorithms implemented in GAP
dealing with permutation groups and character tables, e. g., to find subgroup fusions. As usual, we denote characters by their degrees, supplemented by subscripts
if necessary. Note that we count each pair of faithful characters in 3. Suz only once.
We would like to remark that it is possible to give a complete but more complicated lengthy proof avoiding the module theoretic methods, which uses results for
2. Suz:2 instead. It can be found in [7]. Furthermore, as our results describe all
of the irreducible Brauer characters of the triple cover 3. Suz, the decomposition
numbers for 3. Suz are easily found using the given results; the details are omitted. Especially, while the principal block of 3. Suz exactly contains the irreducible
Brauer characters belonging to the principal block of Suz, it also contains faithful
ordinary characters besides the ordinary characters belonging to the principal block
of Suz. It can be shown that the decomposition matrix for the principal block of
3. Suz in fact is of wedge shape.
Acknowledgements
We wish to thank Lehrstuhl D für Mathematik, RWTH Aachen and IWR, Universität Heidelberg for providing the necessary computer equipment and Deutsche
Forschungsgemeinschaft for financial support in the framework of the research
project ‘Algorithmic Algebra and Number Theory’, which this note is a contribution to. We also wish to thank M. Schönert whose Todd-Coxeter coset enumeration
program we have been allowed to use.
SUZ MOD 3
3
Table 1. Decomposition matrix for the principal block
1
143
364
780
1001
3432
50051
50052
5940
10725
12012
14300
150151
150152
250251
250252
250253
40040
500501
500502
54054
64064
643501
643502
66560
75075
79872
100100
133056
146432
163800
168960
193050
197120
1
1
.
1
2
2
.
.
2
.
.
2
2
2
3
.
.
5
3
3
.
3
5
5
2
5
4
4
3
4
7
6
5
8
.
1
.
1
1
2
1
1
1
1
.
1
1
1
2
1
1
4
3
3
.
4
4
4
2
5
4
4
6
6
8
8
7
8
.
1
1
.
.
.
1
1
1
1
2
1
1
1
1
.
.
3
1
1
2
3
2
2
1
3
3
4
2
3
6
4
5
6
.
.
1
1
1
1
.
.
4
1
2
1
2
2
2
.
.
5
2
2
.
4
3
3
2
4
4
4
2
3
7
5
6
7
.
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1
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1
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1
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1
1
1
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1
1
1
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1
1
1
2
1
2
1
1
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2
2
2
2
.
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1
1
.
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2
1
1
1
1
1
1
1
1
3
2
2
.
3
3
3
2
4
3
2
4
4
5
5
6
6
.
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1
1
1
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1
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1
1
1
1
1
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1
1
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1
1
4
3
3
3
4
3
.
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1
1
1
1
2
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1
1
2
1
1
1
3
1
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2
1
2
4
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5
2
6
5
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1
1
1
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1
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1
1
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2
2
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3
3
4
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1
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1
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1
2. The Results
2.1. The Principal Block. The principal block consists of 34 ordinary and 13
Brauer characters. Its decomposition matrix is given in Table 1. The Brauer
characters are given as follows.
{1, 64, 78, 286, 429, 649, 1938, 2925, 4785, 8436, 14730, 19449, 32967}
2.2. The Second Block. The second block consists of 6 ordinary characters and 5
Brauer characters. Its decomposition matrix can be found in Table 2. The Brauer
4
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 2. Decomposition matrix for the second block
15795
88452
935551
935552
243243
248832
1
1
1
1
.
1
. . . .
1 . . .
1 1 . .
1 . 1 .
1 1 1 1
1 . . 1
Table 3. Decomposition matrix for the third block
18954 1 .
189540 . 1
208494 1 1
characters are given as follows.
{15795, 72657, 51031 , 51032 , 160380}
2.3. The Third Block. The third block consists of 3 ordinary characters, namely
18954, 189540, and 208494, it is of cyclic defect. The decomposition matrix can be
found in [6], we state it for convenience in Table 3.
3. Modular Tables For Some Maximal Subgroups
We make use of the ordinary character tables of four of the maximal subgroups of
. U (2), and 24+6 :3A , and of the 3-modular Brauer
Suz, namely G2 (4), U5 (2), 21+6
4
6
−
character tables of G2 (4), U5 (2), U4 (2), and A6 . As it is not a priori clear that
the subgroup fusions of these subgroups into Suz provided by GAP are compatible
with each other and with the Brauer character tables, we have to check this. It
turns out that there are only ambiguities concerning conjugacy classes of 3-singular
elements and additionally one concerning classes of elements of order 13 in G2 (4).
Hence we are free to choose the subgroup fusions as they are provided by GAP.
Let us now fix some notation. The irreducible Brauer characters of the subgroups
mentioned above are denoted by ϕi , where we follow the enumeration given in [8]
and GAP. The ordinary character of the projective cover of such an irreducible
Brauer character ϕ is denoted by Φ(ϕ). We then obtain projective characters
Φ ↑Suz of Suz by induction from the projective indecomposable characters Φ of
these subgroups. The induced projective characters are denoted by small letter φ’s,
supplemented by a suitable index. Later on, we restrict the φ’s to the respective
blocks being under consideration.
3.1. G2 (4). Let {φ1 , . . . , φ17 } be the following induced characters
Φ(ϕ13 ) ↑Suz , Φ(ϕ7 ) ↑Suz , Φ(ϕ4 ) ↑Suz , Φ(ϕ5 ) ↑Suz , Φ(ϕ3 ) ↑Suz , Φ(ϕ2 ) ↑Suz ,
Φ(ϕ1 ) ↑Suz , Φ(ϕ15 ) ↑Suz , Φ(ϕ8 ) ↑Suz , Φ(ϕ9 ) ↑Suz , Φ(ϕ16 ) ↑Suz , Φ(ϕ10 ) ↑Suz ,
Φ(ϕ17 ) ↑Suz , Φ(ϕ12 ) ↑Suz , Φ(ϕ6 ) ↑Suz , Φ(ϕ18 ) ↑Suz , Φ(ϕ19 ) ↑Suz .
SUZ MOD 3
5
3.2. U5 (2). We take φ18 := Φ(ϕ9 ) ↑Suz .
. U (2). Let ϕ be the the trivial character of 21+6 . U (2). By [4], Lemma
3.3. 21+6
4
1
4
−
−
V.4.3., Φ(ϕ1 ) only consists of characters belonging to the factor group U4 (2), whose
3-modular table is known. As 21+6
is the maximal 2-normal subgroup, the factor
−
. U (2) to U (2) is uniquely determined. This determines Φ(ϕ )
fusion from 21+6
4
4
1
−
and we take φ19 := Φ(ϕ1 ) ↑Suz .
3.4. 24+6 :3A6 . We only need projective indecomposable characters belonging to
blocks which have the maximal 2-normal subgroup 24+6 in their kernel. Hence
again by [4], Lemma V.4.3., we are content with the 3-modular decomposition
matrix of 3. A6 , which in turn is determined by the 3-modular Brauer character
table of A6 . Let φ20 := Φ(ϕ1 ) ↑Suz , φ21 := Φ(ϕ4 ) ↑Suz , φ22 := Φ(ϕ5 ) ↑Suz .
4. The Principal Block
4.1. First of all we form a basic set of Brauer characters for the principal block,
consisting of ordinary characters of 3. Suz restricted to the 3-regular classes. We
choose
BS := {1, 66, 78, 364, 429, 1001, 2145, 29251 , 10725, 14300, 24024, 250252 , 54054}.
This is indeed a basic set of Brauer characters as it can be proved that the restrictions to the 3-regular classes of the ordinary characters belonging to the principal block decompose into BS with integral coefficients. Next we let P S :=
{ψ1 , . . . , ψ13 } where the ψi are the restrictions of
φ16 , φ8 , φ11 , φ18 , φ12 , φ14 , φ9 , φ2 , φ4 , φ3 , φ5 , φ6 , φ7 ,
respectively, to the principal block. This is a basic set of projective characters,
since it can be shown that the elementary divisors of the matrix of scalar products
with BS are all equal to 1.
4.2. In the sequel we denote by BA and P A the bases dual to the current bases
P S and BS with respect to taking scalar products. Then the projective characters
ψ1 , . . . , ψ5 decompose into P A as follows.
ψ1
ψ2
ψ3
ψ4
ψ5
=
=
=
=
=
P A · (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1)t ,
P A · (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0)t ,
P A · (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3)t ,
P A · (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4)t ,
P A · (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 3)t .
Let ψ14 be the restriction of φ1 to the principal block. Then ψ14 decomposes
into P S as ψ14 = ψ5 + 2 · ψ2 − ψ1 . Therefore ψ1 cannot be indecomposable, hence
ψ15 := −ψ1 − ψ2 − ψ3 + ψ4 and ψ16 := ψ1 − ψ15 are projective. We set P S as
follows, the corresponding matrix of scalar products being given in Table 4,
P S := {ψ16 , ψ15 , ψ2 , ψ3 , ψ5 , . . . , ψ13 }.
6
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 4. First matrix of scalar products
























.
. . .
.
. . .
.
. . .
.
. . .
.
. . .
.
. . .
.
. . .
.
. . .
.
. . .
.
. . 1
.
. 1 .
. 1 1 .
1
. . 3
16 15 2 3
.
.
.
.
.
.
.
.
1
1
.
.
3
5
.
.
.
.
.
.
.
1
1
.
2
1
4
6
.
.
.
.
.
.
1
.
1
.
1
1
.
7
.
.
.
.
.
1
.
.
1
3
3
1
2
8

1
.
.
.
. 1
.
.
.
1
2
66 

.
.
1
.
.
78 

. 1
1
.
.
364 

1
.
. 2
1
429 

. 1
. 1
2
1001 

1
. 1
2
1
2145 

.
.
.
.
. 29251 

1
1
1
1
. 10725 

1
2
1
1
2 14300 

2
3
2
2 1 24024 

2
.
. 1
. 250252 

1
1
2
.
. 54054 
9 10 11 12 13
ψi
Table 5. Second matrix of scalar products
























.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. . 1
.
.
. 1
1
.
. 1 .
.
. 1
. .
.
1
.
. 3
2
16 15 18 3 17
.
.
.
.
.
.
.
1
1
.
2
1
4
6
.
.
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.
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.
1
.
1
.
1
1
.
7
.
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.
1
.
.
1
3
3
1
2
8

1
.
.
.
.
1
.
.
.
1
1
65 

.
.
1
.
.
78 

. 1
.
.
.
286 

1
.
. 2
1
429 

. 1
. 1
1
1000 

1
.
. 1
.
2002 

.
.
.
.
. 29251 

1
1
1
1
. 10725 

1
2
1
1
2 14300 

2
3
2
2
1 24024 

2
.
. 1
. 250252 

1
1
2
.
. 54054 
9 10 11 12 13
ψi
4.3. Now we have the following relation ψ14 = −ψ16 − ψ15 + 2 · ψ2 + ψ5 , hence
ψ17 := ψ5 − ψ16 and ψ18 := ψ2 − ψ15 are projective.
From 143 = −1 + 66 + 78 on 3-regular classes we deduce that 65 := 66 − 1 is a
Brauer character. From 1716 = −78 + 364 + 429 + 1001 we find 286 := 364 − 78,
and from 23100 = −1 + 1001 − 29251 + 250252 we find 1000 := 1001 − 1 as Brauer
characters. Using 50051 = 1 − 66 + 2145 + 29251 and 3432 = 1 − 66 − 78 + 429 +
1001 + 2145 we finally get the existence of 2002 := 2145 − 78 − 65. This gives us
new basic sets, whose matrix of scalar products is given in Table 5,
P S := {ψ16 , ψ15 , ψ18 , ψ3 , ψ17 , ψ6 , . . . , ψ13 },
BS := {1, 65, 78, 286, 429, 1000, 2002, 29251 , 10725, 14300, 24024, 250252 , 54054}.
SUZ MOD 3
7
Table 6. Third matrix of scalar products
























.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. . 1
.
.
. 1
1
.
. 1 .
.
. 1
. .
.
1
.
. 3
2
16 15 18 3 17
.
.
.
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.
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.
1
1
.
1
.
4
6
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1
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1
1
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7
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. 1
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1
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1
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. 1
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. 1
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. 2
1
1
. 1
. 1
1
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1
. 1
1
.
.
3
1
2
1
1
2
3
. 3
1
2
1
1
.
.
. 1
.
2
1
1
2
.
.
8 19 10 11 12 13
1
64
78
286
429
1000
1938
29251
8723
14300
21021
22100
54054
ψi
























4.4. Now 12012 decomposes into BS as 12012 = 78 + 286 − 2002 + 29251 + 10725.
As 78, 286 and 29251 are irreducible Brauer characters and cannot be contained
in 2002 we get that 8723 := 10725 − 2002 is a Brauer character. Next 21450 =
−78 + 429 − 29251 + 24024 proves the existence of 21021 := 24024 − 78 − 29251 .
Using 23100 = 1000 − 29251 + 250252 we see that 22100 := 250252 − 29251 is a
Brauer character.
4.5. Now we consider the 12-dimensional self-dual representation of 2. Suz over
GF (3), which is irreducible, since the same is true for its restriction to 2. G (4).
2
The tensor product 12⊗12 is also self-dual and, using the Benson-Carlson Theorem
[2], see also [1], 3.1.9., each direct summand has a dimension divisible by 3. The
tensor product decomposes into its symmetric and antisymmetric parts as 12⊗12 =
12(2+) ⊕ 12(2−) , where 12(2−) = 65 + 1 as Brauer characters. By looking at the
matrix of scalar products we see that either 65 is irreducible or has to be decomposed
into 64 + 1. Hence 12(2−) is uniserial with ascending composition series 1, 64, 1, and
64 := 65 − 1 is a Brauer character.
Furthermore we have 64(2−) = −64 + 78 + 2002, this proves the existence of
1938 := 2002 − 64. We obtain the following basic set of Brauer characters
BS := {1, 64, 78, 286, 429, 1000, 1938, 29251 , 8723, 14300, 21021, 22100, 54054}.
Let ψ19 be the restriction of φ22 to the principal block. The following can be shown
to be a basic set using the matrix of scalar products given in Table 6,
P S := {ψ16 , ψ15 , ψ18 , ψ3 , ψ17 , ψ6 , ψ7 , ψ8 , ψ19 , ψ10 , ψ11 , ψ12 , ψ13 }.
4.6. We consider the restrictions ψ20 and ψ21 of φ20 and φ21 , respectively, to the
principal block. They fulfill the following relations ψ20 = −ψ19 + ψ10 + ψ13 and
ψ21 = ψ8 − 2 · ψ19 + ψ12 . Hence the Brauer character 429 must be irreducible.
Furthermore, ψ20 proves that ψ19 cannot be indecomposable as ψ13 − ψ19 is not
projective. Therefore, ψ19 has to be decomposed. The decompositions
P A · ((0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0) + (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1))t and
8
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 7. Fourth matrix of scalar products
























.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 1
.
.
. 1
1
.
. 1
.
.
. 1
.
.
.
1
.
. 1
2
16 15 18 23 17
.
.
.
.
.
.
.
1
1
.
1
.
4
6
.
.
.
.
.
.
1
.
.
.
1
1
.
7
.
.
.
.
. 1
.
.
.
.
1
.
.
.
.
1
.
.
.
.
1
.
.
.
. 1
.
. 2
1
1
. 1
.
. 1
.
.
.
.
.
.
.
.
.
.
.
.
1
. 1
1
.
.
3
. 2
1
1
1
3
. 3
1
2
1
1
.
.
. 1
.
2
. 1
2
.
.
8 22 10 11 12 13
1
64
78
286
429
936
1938
29251
8723
14299
21021
22100
54054
ψi
























P A · (((0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1) + (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0))t
cannot be correct because of ψ21 . Therefore we have found
ψ22 := P A · (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)t and
ψ23 := P A · (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1)t
as new projective characters.
Because of 5940 = −64+78+2·286+429+2·1000+29251 we have 936 := 1000−64
as a Brauer character. Similarly, using the irreducibility of 429 and the relation
66560 = −2 · 1 − 2 · 64 − 78 + 2 · 429 − 1000 − 1938 + 2 · 29251 − 8723 + 2 · 14300 +
21021 + 22100, we find 14299 := 14300 − 1 to be a new Brauer character. This leads
to new basic sets with matrix of scalar products in Table 7,
P S := {ψ16 , ψ15 , ψ18 , ψ23 , ψ17 , ψ6 , ψ7 , ψ8 , ψ22 , ψ10 , ψ11 , ψ12 , ψ13 },
BS := {1, 64, 78, 286, 429, 936, 1938, 29251 , 8723, 14299, 21021, 22100, 54054}.
4.7. Now ψ20 = −ψ23 − ψ22 + ψ10 + ψ13 proves that ψ24 := ψ13 − ψ22 is projective.
ψ21 = −2·ψ23 +ψ8 −2·ψ22 +ψ12 shows that ψ25 := ψ12 −2·ψ22 and ψ26 := ψ8 −ψ23
are projective characters.
Since 8723 − 936 − 78 + 1 is the sum of two Brauer atoms, one of which equals the
irreducible Brauer character 29251 , the relation 250251 = 1−64+429−1938−8723+
14299 + 21021 proves that 5798 := 8723 − 29251 and 16158 := 21021 − 1938 − 29251
are Brauer characters. Changing basic sets leads to
P S := {ψ16 , ψ15 , ψ18 , ψ23 , ψ17 , ψ6 , ψ7 , ψ26 , ψ22 , ψ10 , ψ11 , ψ25 , ψ24 },
BS := {1, 64, 78, 286, 429, 936, 1938, 29251 , 5798, 14299, 16158, 22100, 54054}.
SUZ MOD 3
9
Table 8. Fifth matrix of scalar products
























.
.
.
.
. . .
.
.
.
.
. 1
.
.
.
.
. . .
.
.
.
. 1
.
.
.
.
.
. . .
.
.
.
1
.
.
.
.
.
.
. . .
.
.
1
.
.
.
.
.
.
.
. . .
. 1
.
.
.
.
.
.
.
.
. . . 1
. 1
.
. 1
.
.
.
.
. . 1
.
.
.
.
.
.
.
.
.
.
. 1 .
.
.
.
.
.
.
.
.
.
. 1 . . 1
. 1
1
.
.
.
.
. 1
1 . . 2
. 2
1
1
1
.
. 1
.
. . . 3
. 1
.
.
.
. 1
.
.
. . 1
1
.
.
. 1
.
1
.
.
. 1 4 .
.
.
. 1
.
.
16 15 18 23 17 6 7 26 22 10 11 25 24
1
64
78
286
429
936
1938
29251
5798
14299
15379
22100
40755
ψi
























4.8. We now consider some tensor products between Brauer characters. We have
64 ⊗ 286 = −1 − 2 · 64 − 78 − 2 · 286 + 29251 + 16158. Therefore 15379 := 16158 −
1 − 2 · 64 − 78 − 2 · 286 is a Brauer character, as 29251 is irreducible. Now let
40755 := 286(2−) = 64 + 936 − 14299 + 54054. We change BS again, obtaining the
new matrix of scalar products in Table 8,
BS := {1, 64, 78, 286, 429, 936, 1938, 29251 , 5798, 14299, 15379, 22100, 40755}.
4.9. Considering 54054 = −64 − 936 + 14299 + 40755, we find that 13299 :=
14299 − 936 − 64 is a Brauer character.
From ψ20 = −ψ23 + ψ10 + ψ24 we see that ψ27 := ψ10 − ψ23 is projective, and
from ψ21 = −ψ23 + ψ26 + ψ25 we conclude that the character ψ28 := ψ26 − ψ23 is
projective.
Using the relation 250251 = 2·1+2·64+78+2·286+429+936−5798+13299+15379
and the decomposition 13299 = (0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0) · BA, it can be shown
that 4863 := 5798 − 936 + 1 and 8514 := 13299 − 4863 + 78 are Brauer characters.
Changing basic sets gives, with corresponding matrix of scalar products in Table 9,
P S := {ψ16 , ψ15 , ψ18 , ψ23 , ψ17 , ψ6 , ψ7 , ψ28 , ψ22 , ψ27 , ψ11 , ψ25 , ψ24 },
BS := {1, 64, 78, 286, 429, 936, 1938, 29251 , 4863, 8514, 15379, 22100, 40755}.
4.10. Using 5798 = −1 + 936 + 4863 we get that 935 := 936 − 1 is a Brauer
character. 60060 = 3 · 1 + 2 · 64 − 2 · 78 + 286 + 429 + 4863 + 2 · 8514 + 15379 + 22100
proves that 8436 := 8514 − 78 is a Brauer character. Since we find 64 ⊗ 935 =
2 · 1 + 2 · 286 − 935 − 1938 + 29251 + 4863 + 2 · 8436 + 15379 + 22100 we get the
existence of 20162 := 22100 − 1938. We now have the following basic set,
BS := {1, 64, 78, 286, 429, 935, 1938, 29251 , 4863, 8436, 15379, 20162, 40755}.
10
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 9. Sixth matrix of scalar products

























1
.
.
.
.
. . .
.
.
.
.
. 1
.
.
.
.
. . .
.
.
.
. 1
.
64 

.
.
.
.
. . .
.
.
.
1
.
.
78 

.
.
.
.
. . .
.
.
1
.
.
.
286 

.
.
.
.
. . .
. 1
.
.
.
.
429 

.
.
.
.
. . . 1
. 1
.
. 1
936 

.
.
.
.
. . 1
.
.
.
.
.
. 1938 

.
.
.
.
. 1 .
.
.
.
.
.
. 29251 

.
.
.
. 1 . .
.
.
. 1
.
. 4863 

.
.
. 1
. . .
.
.
. 1
.
. 8514 

.
. 1
.
. . . 3
. 1
.
.
. 15379 

. 1
.
.
. . 1
1
.
.
. 1
. 22100 

1
.
.
. 1 4 .
.
.
. 1
.
. 40755 
16 15 18 23 17 6 7 28 22 27 11 25 24
ψi
Table 10. Seventh matrix of scalar products

























1
.
.
.
.
. . .
.
.
.
.
. 1
.
.
.
.
. . .
.
.
.
. 1
.
64 

.
.
.
.
. . .
.
.
.
1
.
.
78 

.
.
.
.
. . .
.
.
1
.
.
.
286 

.
.
.
.
. . .
. 1
.
.
.
.
429 

.
.
.
.
. . . 1
.
.
.
.
.
649 

.
.
.
.
. . 1
.
.
.
.
.
. 1938 

.
.
.
.
. 1 .
.
.
.
.
.
. 29251 

.
.
.
. 1 . .
.
.
. 1
.
. 4863 

.
.
. 1
. . .
.
.
.
.
.
.
8436 

.
. 1
.
. . . 3
. 1
.
.
. 15379 

. 1
.
.
. . . 1
.
.
. 1
. 20162 

1
.
.
. 1 4 .
.
.
. 1
.
. 40755 
16 15 18 23 17 6 7 28 22 27 11 25 24
ψi
4.11. The only remaining question concerning 935 is, whether it is irreducible or
649 := 935 − 286 is an irreducible Brauer character. In Section 8.1 we show that
649 exists. We replace the basic set, getting the scalar products in Table 10,
BS := {1, 64, 78, 286, 429, 649, 1938, 29251 , 4863, 8436, 15379, 20162, 40755}.
4.12. Now assume that 4863 is irreducible. Using the relations 78 ⊗ 649 = −649 −
4863 + 15379 + 40755 and 649(2−) = −78 + 1938 − 29251 + 40755 + ϑ, where ϑ
denotes the component of 649(2−) belonging to the non-principal blocks, we see
that 40755 contains 4863 and 78. This is a contradiction to the decomposition of
40755 into atoms as can be seen from the matrix of scalar products given in Table
10. Therefore 4785 := 4863 − 78 exists.
SUZ MOD 3
11
Table 11. Eighth matrix of scalar products

























1
.
.
.
.
. . .
.
.
.
.
. 1
.
.
.
.
. . .
.
.
.
. 1
.
64 

.
.
.
.
. . .
.
.
.
1
.
.
78 

.
.
.
.
. . .
.
.
1
.
.
.
286 

.
.
.
.
. . .
. 1
.
.
.
.
429 

.
.
.
.
. . . 1
.
.
.
.
.
649 

.
.
.
.
. . 1
.
.
.
.
.
. 1938 

.
.
.
.
. 1 .
.
.
.
.
.
. 29251 

.
.
.
. 1 . .
.
.
.
.
.
.
4785 

.
.
. 1
. . .
.
.
.
.
.
.
8436 

.
. 1
.
. . . 2
. 1
.
.
. 14730 

. 1
.
.
. . .
.
.
.
.
1
. 19513 

1
.
.
.
. 3 .
.
.
.
.
.
. 32967 
16 15 18 23 17 6 7 28 22 27 11 25 24
ψi
Then we get that 32967 := 40755 − 78 − 29251 − 4785 is a Brauer character.
Also due to 78 ⊗ 649 we conclude that 14730 := 15379 − 649 is a Brauer character.
When considering 64 ⊗ 649 = 2 · 1 + 78 + 286 − 649 + 4785 + 2 · 8436 + 20162 we
conclude that 19513 := 20162 − 649 is a Brauer character. We take the new basic
set as follows, with matrix of scalar products as it is in Table 11,
BS := {1, 64, 78, 286, 429, 649, 1938, 29251 , 4785, 8436, 14730, 19513, 32967}.
4.13. Hence we know all the irreducible Brauer characters in the principal block
except ϕ11 , ϕ12 and ϕ13 . There are the following possible cases left. ϕ11 = 14730 −
x · 286 − y · 649 where x ∈ {0, 1} and y ∈ {0, 1, 2}, ϕ12 = 19513 − z · 64 where
z ∈ {0, 1}, and ϕ13 = 32967 − w · 29251 where w ∈ {0, 1, 2, 3}. The values for x, y,
z, and w are determined in Sections 8.2, 8.3, and 8.5.
5. The Second Block
5.1. Let {ψ29 , . . . , ψ33 } be the restrictions of φ15 , 13 φ14 , φ9 , φ16 , and 13 φ5 , respectively, to the second block.
Here we use the following notation. Inducing the projective character φ14 up to
Suz and restricting it to the second block yields a character whose scalar products
with all the ordinary characters in this block are divisible by 3. Hence we can divide
all these by 3 and still have a projective character, which then is denoted by ψ30 .
The character ψ33 is found similarly.
Then we get ψ30 + ψ31 = 2 · (88452 + 935551 + 935552 + 4 · 243243 + 4 · 248832)
as ordinary characters. Hence ψ34 := (ψ30 + ψ31 )/2 is projective. Now we take
P S := {ψ29 , ψ30 , ψ34 , ψ32 , ψ33 }
and the following ordinary characters restricted to the 3-regular classes as BS. By
the matrix of scalar products in Table 12 these are indeed basic sets,
BS := {15795, 88452, 160380, 51031 , 51032 }.
12
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 12. First matrix of scalar products









.
.
.
. 1 15795
. 1
1
1
1 88452 

. 2
3
4
1 160380 

1
.
.
.
.
51031 

1
.
. 1
.
51032 
29 30 34 32 33
ψi
Table 13. Second matrix of scalar products








.
.
.
. 1 15795
.
.
. 1
1 88452
.
. 1
.
.
51031
. 1
.
.
.
51032
1
.
.
. 1 160380
41 38 39 40 33
ψi








5.2. Now let ψ35 , ψ36 , and ψ37 be the restrictions of φ17 , φ13 , and φ10 , respectively,
to the second block. These decompose into P S as ψ35 = ψ29 − 2 · ψ30 + 4 · ψ34 − ψ32 ,
ψ36 = 9 · ψ30 − 5 · ψ34 , and ψ37 = −ψ30 + 2 · ψ34 .
We now show that ψ30 decomposes into projective indecomposable characters as
ψ30 = P A · ((0, 1, 0, 0, 0) + (0, 0, 1, 0, 0) + (0, 0, 1, 0, 0))t . There are three other cases
to exclude. (i) ψ30 is indecomposable. That is impossible due to ψ36 and ψ37 . (ii)
ψ30 = P A · ((0, 1, 0, 0, 0) + (0, 0, 2, 0, 0))t . This is not correct because it implies the
existence of an indecomposable projective character which is twice an atom. (iii)
ψ30 = P A · ((0, 1, 1, 0, 0) + (0, 0, 1, 0, 0))t . As ψ30 + ψ31 = 2 · ψ34 , we have ψ34 =
P A · ((0, 1, 1, 0, 0) + (0, 0, 1, 0, 0) + (0, 0, 1, 0, 0))t , hence ψ36 = P A · (4 · (0, 1, 1, 0, 0) −
1 · (0, 0, 1, 0, 0))t , which is not a positive decomposition into indecomposables.
Furthermore ψ35 proves that ψ29 cannot be indecomposable. Now we put ψ38 :=
P A · (0, 0, 0, 0, 1)t , ψ39 := P A · (0, 0, 0, 1, 0)t , ψ40 := P A · (0, 1, 0, 0, 0)t , and ψ41 :=
P A · (0, 0, 1, 0, 0)t , and let, with matrix of scalar products given in Table 13,
P S := {ψ41 , ψ38 , ψ39 , ψ40 , ψ33 },
BS := {15795, 88452, 51031 , 51032 , 160380}.
5.3. Finally, let ψ42 be the restriction of φ19 to the second block. Then ψ42 =
−ψ41 + ψ40 + 5 · ψ33 . Therefore ψ43 := ψ33 − ψ41 is projective. Using the relation
243243 = −15795 + 88452 + 51031 + 51032 + 160380 we see that 72657 := 88452 −
15795 is a Brauer character. This concludes the proof for the second block.
6. Some Remarks On Condensation
To complete the decomposition matrix for the principal block, we construct and
analyze several matrix and permutation representations of Suz explicitly. We use
fixed point condensation to obtain modules of a size which makes it possible to
SUZ MOD 3
13
examine submodule structures explicitly. In this section we give a few comments
and remarks concerning this method and how it is applied.
6.1. Let F be a field, H a subgroup of a finite group G, such that the characteristic
of F does not divide the order |H|. Then we have the following idempotent in the
group algebra F G,
X
e = eH := |H|−1 ·
h.
h∈H
Now let V be an F G-module. Then the subset V e of V is the set of fixed points
of V under the action of H. As V is semisimple as a module for F H, the dimension
of V e can be determined by calculating the ordinary scalar product between the
restriction to H of the Brauer character of V and the trivial character of H. V e is
a module for the Hecke algebra eF Ge. Hence we have a condensation functor from
the category of finitely generated F G-modules to the category of finitely generated
eF Ge-modules. The application of this functor to an F G-module V is called fixed
point condensation. A few of the properties of this functor, especially as far as
the submodule structures of V and V e are concerned, are described in [15] or [12].
Especially, if we have Se 6= {0} for each constituent S of V , then the submodule
lattices of V and V e are naturally isomorphic. The condensation functor can be
efficiently computed for permutation modules of degree up to several million and
for tensor product modules of dimension up to several ten thousand. Programs
performing these tasks are implemented in the MeatAxe and in the TensorCondense
package, which both have already been mentioned in Section 1.
6.2. Now let {g1 , g2 , . . . , gn } be a set of generators for G. Then the subalgebra
C of eF Ge generated by {eg1 e, eg2 e, . . . , egn e} is called the condensation algebra
with respect to this generating set. It is not necessarily equal to eF Ge. Hence we
have to distinguish carefully between V e as a C-module and as an eF Ge-module.
We analyze the submodule structure of V e as a C-module using the computational
methods mentioned above, and then we have to draw conclusions about its structure
as an eF Ge-module. One example how this works is given in Remark 6.3 below,
which is applied in 8.3. A more systematic approach to these problems can be
found in [7].
6.3. Remark. For later use, we briefly recall the following notion from [10]. Let
R be a ring. An R-module is called local, if it has a uniquely determined maximal
R-submodule. A local R-module is called S-local, if its head is isomorphic to the
simple R-module S.
Now let M be an eF Ge-module. Let U 0 < U be eF Ge-submodules of M , such
that S := U/U 0 is an irreducible eF Ge-module, and assume further that S̃ := S|C
is an irreducible C-module. Then there is an S̃-local C-submodule L̃ ≤ U , such
that L̃ · eF Ge ≤ U is an S-local eF Ge-submodule.
To see this, note that by the assumptions there is an S-local eF Ge-submodule
L ≤ U having the maximal eF Ge-submodule L0 < L, such that L + U 0 = U and
L ∩ U 0 = L0 . As S|C is irreducible, there is an S̃-local C-submodule L̃ ≤ L having
the maximal C-submodule L̃0 < L̃, such that L̃ + L0 = L and L̃ ∩ L0 = L0 . Now we
have L̃ · eF Ge = L.
14
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 14. Permutation representations
Suz.p1782
G2 (4)
1782 1 + 780 + 1001
Suz.p22880 3. U4 (3):2
22880 1 + 364 + 780 + 5940 + 15795
. U (2) 135135 1 + 143 + 780 + 3432 + 5940+
Suz.p135135 21+6
4
−
14300 + 18954 + 250251 + 66560
6.4. One of the most important tools to ensure the existence of certain submodules
of V e as an eF Ge-module is the following theorem. First we have to recall the
following notion, which together with a few different characterisations can be found
in [9], Definition I.17.1.
Let (K, R, F ) be a modular system for G, i. e., R is a discrete valuation ring
with fraction field K and quotient field F , such that K and F are splitting fields
for G. Let M be a free R-module. An R-submodule N ≤ M is called R-pure, if
the quotient M/N is again free as an R-module. In this case, it follows that the
F -modular reduction of N is a submodule of the F -modular reduction of M .
6.5. Theorem. (Zassenhaus and others). Let (K, R, F ) be as above and let
M be an R-free RG-module with ordinary character χ, such that χ = χ0 + χ00 as
ordinary characters. Then there exists an R-pure RG-submodule N ≤ M with
character χ0 .
For a proof we refer to [9], Theorem I.17.3. As the F -modular reduction of N is a
submodule of the F -modular reduction of M , this theorem ensures the existence of
F G-submodules of M whose modular constituents can be controlled.
7. Construction Of Representations
7.1. We now give constructions for three transitive permutation representations,
Suz.p1782, Suz.p22880, and Suz.p135135, we use in the sequel. In Table 14 we
write down the corresponding one point stabilizer, the degree, and the permutation
character.
7.2. We start with the presentation Suz:2 := ha, b, c, d, e, f, g|Ri, where R is the
following set of relators:
{a2 , b2 , (ab)5 , c2 , (ac)2 , (bc)3 , d2 , (ad)2 , (bd)2 , (cd)8 , e2 , (ae)2 , (be)2 , (ce)2 , (de)3 ,
f 2 , (af )2 , (bf )3 , (cf )2 , (df )2 , (ef )2 , g 2 , (ag)2 , (bg)2 , (cg)2 , (dg)2 , (eg)2 , (f g)8 ,
a−1 (cd)4 , (abcdecbdc)7 , (abcf )5 , a−1 (f g)4 , (bf gf g)5 }.
This is the second presentation to be found in [3], p. 131.
Let A := f and B := gabcf deg. Using a Todd-Coxeter coset enumeration with
respect to the subgroup Suz := hA, Bi of Suz:2, we find that this subgroup has
indeed index 2. Using a Todd-Coxeter coset enumeration again, now with respect
to the subgroup ha, b, c, d, e, f, h2 gi, where h := gf bcdcbf , gives us a permutation
representation of Suz:2 on 1782 points. By restriction to Suz we obtain the representation Suz.p1782. Similarly, using the subgroup ha, b, c, d, f, gi and restriction
to Suz we obtain the permutation representation Suz.p22880.
By some abuse of notation we also denote the homomorphic images of A and
B in several representations by the same symbols. This should not lead to any
confusion.
SUZ MOD 3
15
7.3. Our next task is to construct the permutation representation Suz.p135135.
. U (2), which is
The one point stabilizer is the forth maximal subgroup M4 := 21+6
4
−
the centralizer of a 2A element of Suz. Note that we have already encountered M4
in Section 3.3. Now A is of order 2, and as it has 54 fixed points in the representation
Suz.p1782, it is a 2A element. Let C1 := AB, C2 := C1 B, C3 := C2 B, and
D1 := (A(C2 C3 C22 C32 )5 )4 , D4 := (A(B 2 C32 C2 C1 )5 )2 ,
D2 := (A(B 3 C1 )5 )4 ,
D5 := (A(((C1 C2 )2 C2 )3 C1 C22 )5 )5 ,
5
6 6
D3 := (A((C1 C2 ) C2 ) ) , D6 := (A(B 2 (C3 C2 C3 )2 C2 C3 (C3 C2 C3 )3 C2 C1 )5 )4 .
It is then shown, again using GAP, that hD1 , D2 , D3 , D4 , D5 , D6 i is a subgroup of
the centralizer of A in Suz, having order 3317760, hence M4 is generated by these
elements.
Next we consider the matrix representation corresponding to Suz.p1782. Its
entries are rational integers, hence they can be reduced modulo the rational prime
2, giving a matrix representation of Suz over GF (2). Now the MeatAxe is used to
‘chop’ this module. Especially, we find a constituent 142GF (2) of dimension 142.
Considering the restriction of 142GF (2) to M4 , the MeatAxe shows that there is
exactly one submodule isomorphic to the trivial M4 -module. Clearly, the orbit of
this vector under the action of Suz has length 135135, which gives us Suz.p135135.
7.4. Our next task is to find a suitable condensation subgroup, which in our case
has to be a group of order prime to 3. The maximal normal subgroup O2 := 21+6
−
of M4 , an extraspecial group of order 128, turns out to be suitable for our purposes.
We define E1 := D1 D2 D3 , E2 := D4 D5 , F1 := E1 E2 , F2 := F1 E2 , F3 := F2 E2 ,
F4 := E1 F1 , and
G1 := (F22 F3 )2 ,
G3 := (F42 F1 F43 F1 )2 , G5 := (F42 F1 F43 F1 F4 F1 )6 ,
2 2
2
G2 := (F2 F3 F2 F3 ) , G4 := (F44 F1 )6 ,
G6 := (F1 F2 (F1 F22 )2 (F1 F2 )3 F2 )6 .
It is seen using GAP that we have O2 = hG1 , G2 , G3 , G4 , G5 , G6 i.
In the sequel let e := eO2 denote the idempotent corresponding to O2 . We consider condensed modules as modules for the algebra generated by {eXe, eY e, eZe},
where X := C12 C2 , Y := C14 C2 , and Z := C1 C2 C12 C22 . This algebra, which in the
following is referred to as the condensation algebra, is contained in eF Ge, but of
course we cannot be sure that it is equal to it.
7.5. Finally, we determine the dimension of the condensed modules corresponding
to the ordinary characters of Suz up to degree 66560, by computing the scalar
products of the restrictions of these characters to the subgroup O2 with the trivial
character of O2 . Using GAP, we find that O2 contains 55 elements of order 2 and 72
elements of order 4. Using the fusion of M4 into Suz determined in Section 3 and
the factor fusion from Section 3.3, the scalar products can be computed, the result
is given in Table 15. From these data we compute the dimensions of the condensed
modules corresponding to the Brauer characters in BS given in Section 4.12, which
is used from Section 8.2 on. They can be found in Table 16. Note that in Section
8.1 we use BS given in Section 4.10 instead. The condensed module corresponding
to the Brauer character 935 in there has dimension 19.
8. Analyzing Representations
8.1. Suz.p1782. The aim of this section is to show the existence of 649, as was asserted in Section 4.11, and to find the isomorphism types of the condensed modules
16
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 15. Ordinary characters and the dimensions of the corresponding condensed modules
1
1
143 16
364 24
780 27
1001 26
3432 48
50051
70
50052
70
5940 135
10725
12012
14300
150151
150152
15795
18954
250251
250252
180
234
195
216
216
243
162
215
100
250253
40040
500501
500502
54054
64064
643501
643502
66560
100
490
315
315
612
564
510
510
520
Table 16. Brauer characters and the dimensions of the corresponding condensed modules
429 7
649 5
1938 10
2925 45
4785 91
1 1
64 5
78 10
286 14
8436 68
14730 84
19513 40
32967 297
Table 17. Constituents of the condensed representation Suz.p1782
1
780
1
1
64
286
429
1001
1
64
935
1
1a
1
1a
1
5a
1
14a
1
7a
2
1a
1
5a
1 5b + 14a
corresponding to the 3-modular constituents of Suz.p1782. The decomposition of
the ordinary constituents of Suz.p1782 into BS given in Section 4.10 is shown in Table 17. In the first column we give the ordinary constituents and in the second and
third columns their 3-modular constituents and the corresponding multiplicities.
To show the existence of 649, we consider the matrix representation corresponding to Suz.p1782. Its entries are rational integers, hence they can be reduced
modulo the rational prime 3. Then this module is ‘chopped’, giving 649 explicitly as one of the constituents. The explicit matrices are needed in Section 8.5.
We remark that it is possible to show the mere existence of 649 by analyzing the
submodule lattice of the condensed module corresponding to Suz.p1782, without
‘chopping’ the permutation module explicitly.
We now use the MeatAxe to construct and ‘chop’ the condensed module corresponding to Suz.p1782 with respect to the subgroup O2 . It is found to have
SUZ MOD 3
17
dimension 54 and its constituents as a module for the condensation algebra are
1a4 , 5a2 , 5b, 7a, 14a2 , where the exponents denote multiplicities. Counting multiplicities and dimensions of these constituents, we find that the constituents 1, 64,
286, 429, and 649 condense onto 1a, 5a, 14a, 7a, and 5b, respectively. This correspondence is given in the forth column of Table 17.
8.2. Suz.p135135. We now determine x, y, and z, see Section 4.13, by an analysis
of the condensed module M corresponding to Suz.p135135. The constituents of
M , which has dimension 1319, as a module for the condensation algebra, as they
are found by the MeatAxe, are as follows:
1a14 , 5a10 , 5b7 , 7a6 , 10a5 , 10b2 , 14a11 , 35a1 , 45a3 , 68a4 , 84a2 , 91a2 , 162a1 .
The ordinary constituents of Suz.p135135 and their decomposition into BS given
in Section 4.12 is shown in Table 18. The ordinary constituent 18954 is the only
one not belonging to the principal block.
Comparing dimensions, it follows that 18954 corresponds to the constituent 162a.
Therefore M has a unique direct summand M162 isomorphic to 162a. We use the
MeatAxe to find a peakword, see [10], for 162a. The kernel of its action on M is a
subspace of M162 . It is now a standard application of the MeatAxe to compute the
quotient module Mpb := M/M162 . In the sequel we examine Mpb in more detail.
From the results in Section 8.1 we know that the multiplicity of the constituent
14a in Mpb is greater than or equal to the multiplicity of the irreducible Brauer
character 286 in the decomposition of the permutation character of Suz.p135135.
Now a comparison of multiplicities shows that 14730 cannot contain another 286,
hence we have x = 0. An analogous argument using 5b, which corresponds to 649,
shows that also y = 0. Hence ϕ11 = 14730 is irreducible. We remark that this
result could also be deduced using the condensed tensor product module which is
considered in Section 8.5.
8.3. Now we turn our attention to the determination of z. Comparing dimensions and multiplicities it is straightforward that 91a, 84a, 68a, and 45a correspond
to 4785, 14730, 8436, and 2925, respectively. Hence 35a is a constituent of the
condensed module corresponding to 19513. Now 19513 condenses onto a module
of dimension 40. Counting multiplicities of constituents which have not yet been
shown to correspond to an irreducible Brauer character, we find that 19513 condenses onto a module having the constituents 35a, 5a.
The MeatAxe finds a peakword for 5a. Then each minimal 5a-local submodule
of Mpb is generated by an element of the kernel of its action on Mpb , as follows from
the results in [10]. We now look for those of them which have a dimension less than
44; this bound becomes meaningful, as soon as we have defined the submodule U43 a
few lines below. The MeatAxe shows that there are exactly four such submodules.
One submodule L6 of dimension 6 and three submodules L127 , L227 , and L327 of
dimension 27. Furthermore, each of the L6 and L1,2,3
contain the constituent 5a
27
exactly once, and we have L6 ∩ L127 = L6 ∩ L227 = L6 ∩ L327 , being of dimension 1.
By the Zassenhaus Theorem 6.5, there is an eF Ge-submodule U16 of Mpb of
dimension 16 corresponding to the ordinary constituent 143 of Suz.p135135. Since
all the irreducible 3-modular constituents of 143 are known and 64 is one of them,
U16 has 5a as one of its constituents. Hence by Remark 6.3, it follows that
L̃1 := L6 · eF Ge is a 5a-local eF Ge-submodule of U16 . Again by the Zassenhaus
Theorem, there exists an eF Ge-submodule U43 of dimension 43 which contains U16
18
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
Table 18. Constituents of the condensed representation Suz.p135135
1
143
780
3432
5940
14300
250251
66560
18954
1
1
64
78
1
64
286
429
1
64
286
429
649
1938
1
64
78
286
429
649
2925
1
64
78
286
649
4785
8436
1
64
78
286
429
649
8436
14730
1
64
78
286
429
649
1938
2925
4785
8436
14730
19513
18954
1
1
1
1
1
1
1
1
2
2
1
1
1
1
2
1
1
4
1
2
1
2
1
1
1
1
1
1
3
2
1
2
1
1
1
1
2
1
1
2
2
2
1
2
1
2
1
1
1
1a
1a
5a
10a
1a
5a
14a
7a
1a
5a
14a
7a
5b
10b
1a
5a
10a
14a
7a
5b
45a
1a
5a
10a
14a
5b
91a
68a
1a
5a
10a
14a
7a
5b
68a
84a
1a
5a
10a
14a
7a
5b
10b
45a
91a
68a
84a
5a + 35a
162a
SUZ MOD 3
19
Table 19. Constituents of the condensed representation Suz.p22880
1
78
286
780
1
64
286
429
5940
1
64
78
286
429
649
2925
15795 15795
1
364
1
1a
1 10a
1 14a
1
1a
1
5a
1 14a
1
7a
2
1a
1
5a
1 10a
4 14a
1
7a
2
5b
1 45a
1 243a
and corresponds to the ordinary character 143 + 780. Now the 3-modular constituents of 780 are also known and 64 is again one of them, hence for one of the
Li27 , L127 say, we get L̃2 := L127 ·eF Ge as a 5a-local eF Ge-submodule of U43 . Clearly,
we have L̃1 6= L̃2 .
Using the Zassenhaus Theorem once again, we see that Mpb contains an eF Gesubmodule U520 of dimension 520 which corresponds to the ordinary constituent
66560 of Suz.p135135. The MeatAxe finds a peakword for the constituent 35a. The
kernel of its action on Mpb is of dimension 1, and it generates a 35a-local submodule
L448 of Mpb of dimension 448. Since U520 has a constituent 35a, we have L448 ≤
U520 . Now the MeatAxe shows that L6 +L127 ≤ L448 . Hence we have L̃1 + L̃2 ≤ U520 .
Therefore, the ordinary character 66560 contains the irreducible Brauer character
64 at least twice. Hence we have z = 1 and ϕ12 = 19513 − 64 = 19449 as an
irreducible Brauer character.
8.4. Suz.p22880. Before we finally determine w, see Section 4.13, we have to do
some preparatory work. The ordinary constituents of the permutation character
of Suz.p22880 are given in Table 19. The ordinary character 15795 belongs to the
second block and reduces irreducibly modulo 3, as was shown in Section 5. Hence
all the 3-modular constituents of Suz.p22880 are known.
The condensed module N corresponding to Suz.p22880 has dimension 430. Using the MeatAxe we find its constituents as a module for the condensation algebra:
1a4 , 5a2 , 5b2 , 7a2 , 10a2 , 14a6 , 45a, 243a. Counting dimensions and multiplicities, we
find that 10a is the constituent the irreducible Brauer character 78 condenses to.
Now again by the Zassenhaus Theorem 6.5, there is an eF Ge-submodule U24
of N of dimension 24, which corresponds to the ordinary constituent 364 of the
permutation character. Using the peakword technique mentioned earlier, it is shown
that there is exactly one 10a-local submodule L ≤ N of dimension not exceeding
24. Since L has dimension 24, we have U24 = L. The methods explained in
[10] also yield a generating vector v for L. Now the condensed module V e is a
subset of V . Hence v, considered as an element of V , generates a submodule of
20
CHRISTOPH JANSEN AND JÜRGEN MÜLLER
the permutation module of dimension 364, which is a 3-modular reduction of the
ordinary irreducible module of dimension 364. It is now a standard application
of the MeatAxe to compute the action of the chosen generators for Suz on this
submodule, and to obtain its constituent 78.
8.5. Suz.78 ⊗ 649. Now we are ready to determine w, see Section 4.13, by applying
the TensorCondense package. A short comment on the application of TensorCondense is in order. A crucial step in the procedure is to find suitable symmetry bases,
see [11], for the tensor factors 78 and 649. The package provides two different
methods to find such bases, the peakword method and the γ-operator method, none
of which is well-suited for our case. This is because our condensation subgroup O2
has 64 different linear representations. But in the case of a linear representation the
corresponding part of a suitable symmetry basis can be found by explicitly computing simultaneous eigenspaces for the generators of the condensation subgroup.
This is done by concatenating a few of the standard applications of the MeatAxe.
The Brauer character of the tensor product 78 ⊗ 649 decomposes as 78 ⊗ 649 =
2925 + 14730 + 32967 into BS given in Section 4.12. Hence we show that the
corresponding condensed module has composition length 3 as a module for the
condensation algebra. It then follows that w = 0, i. e., ϕ13 = 32967 is irreducible.
Note that this gives an alternative proof of the irreducibility of ϕ11 = 14730, see
Section 8.2.
Now applying TensorCondense using the explicit matrices for 78 and 649 constructed in Sections 8.4 and 8.1 yields the condensed module, which is of dimension
426. The standard MeatAxe shows that it has the constituents 45a, 84a, 297a as a
module for the condensation algebra, and we are done.
References
[1] D. Benson: Representations and Cohomology, vol. I, Cambridge, 1991.
[2] D. Benson, J. Carlson: Nilpotent Elements in the Green Ring, J. Algebra 104:329–350, 1986.
[3] J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson: Atlas of Finite Groups, Clarendon,
1985.
[4] W. Feit: The representation theory of finite groups, North-Holland, 1982.
[5] G. Hiss, C. Jansen, K. Lux, R. Parker: Computational Modular Character Theory, Preprint,
1992.
[6] G. Hiss, K. Lux: Brauer Trees of Sporadic Groups, Clarendon, 1989.
[7] C. Jansen: Ein Atlas 3-modularer Charaktertafeln, Dissertation, RWTH Aachen, 1995.
[8] C. Jansen, K. Lux, R. Parker, R. Wilson: An Atlas of Brauer Characters, Oxford University
Press, 1995.
[9] P. Landrock: Finite Group Algebras and Their Modules, Cambridge, 1983.
[10] K. Lux, J. Müller, M. Ringe: Peakword Condensation and Submodule Lattices, J. Symb.
Comp. 17:529–544, 1994.
[11] K. Lux, M. Wiegelmann: Condensing Tensor Product Modules, in preparation.
[12] J. Müller: 5-modulare Zerlegungszahlen für die sporadische einfache Gruppe Co3 , Diplomarbeit, RWTH Aachen, 1991.
[13] R. Parker: The Computer Calculation of Modular Characters, in Atkinson (ed.): Computational Group Theory, 1984.
[14] M. Ringe: The C-MeatAxe, Documentation, RWTH Aachen, 1992.
[15] A. Ryba: Computer Condensation of Modular Representations, J. Symb. Comp. 9:591–600,
1990.
[16] M. Schönert (ed.): GAP-3.4 Manual, RWTH Aachen, 1994.
[17] M. Wiegelmann: Fixpunktkondensation von Tensorproduktmoduln, Diplomarbeit, RWTH
Aachen, 1994.
SUZ MOD 3
21
(Christoph Jansen) Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64,
D-52056 Aachen
E-mail address: [email protected]
(Jürgen Müller) Interdisziplinäres Zentrum für wissenschaftliches Rechnen, Universität Heidelberg, Im Neuenheimer Feld 368, D-69120 Heidelberg
E-mail address: [email protected]